Nat.Lab. Unclassified Report NL-UR 2002/823
Date of issue: August 2002
Introduction to and Usage of the
Bipolar Transistor Model Mextram
J.C.J. Paasschens and R. v.d. Toorn
Unclassified report
©
Koninklijke Philips Electronics N.V. 2002
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
Authors’ address data: J.C.J. Paasschens WAY41; Jeroen.Paasschens@philips.com
R. v.d. Toorn WAY41; Ramses.van.der.Toorn@philips.com
©Koninklijke Philips Electronics N.V. 2002
All rights are reserved. Reproduction in whole or in part is
prohibited without the written consent of the copyright owner.
ii
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
August 2002— NL-UR 2002/823
Unclassified Report:
NL-UR 2002/823
Title:
Introduction to and Usage of the Bipolar Transistor Model Mextram
Author(s):
J.C.J. Paasschens and R. v.d. Toorn
Part of project:
Compact modelling
Customer:
Semiconductors
Keywords:
Mextram, spice, equivalent circuits, analogue simulation, semiconductor device models, semiconductor device noise, nonlinear
distortion, heterojunction bipolar transistors, bipolar transistors,
integrated circuit modelling, semiconductor technology
Abstract:
Mextram is a compact model for vertical bipolar transistors. For a
designer such a compact model, together with the parameters for
a specific technology, is the interface between the circuit design
and the real transistor. To be able to use Mextram in an efficient
way one needs an understanding of some of the basics of Mextram. This report is meant to give a desginer an introduction into
Mextram, level 504.
In this report we describe Mextram 504. We discuss its equivalent
circuit and how the various elements of this circuit are connected
to the various regions of a real transistor. We discuss the similarities and differences of Mextram 504 and the well-known SpiceGummel-Poon model. Then we sketch the effects that can play a
role in the low-doped collector epilayer of a bipolar transistor and
how these are modelled within Mextram 504.
We show how one can handle the substrate resistance/capacitance
network, that is not an integral part of Mextram. Furthermore,
we discuss the possibilities of modelling self heating and mutual
heating with Mextram 504. At last we show what information can
be found from the operating point information that is supplied by
many circuit simulators.
©Koninklijke Philips Electronics N.V. 2002
iii
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
Preface
The Mextram bipolar transistor model has been put in the public domain in Januari 1994.
At that time level 503.1 of Mextram was used within Koninklijke Philips Electronics N.V.
In June 1995 level 503.2 was released which contained some improvements.
Mextram level 504 contains a complete review of the Mextram model. This document
gives an introduction for designers.
August 2002, J.P.
History of model
January 1994
June 1995
: Release of Mextram level 503.1
: Release of Mextram level 503.2
– Improved description of Early voltage
– Improved description of cut-off frequency
– Parameter compatible with level 503.1
June 2000
: Release of Mextram level 504 (preliminary version)
– Complete review of the model
April 2001
: Release of Mextram level 504
Small fixes:
– Parameters Rth and Cth added to MULT -scaling
– Expression for α in operating point information fixed
Changes w.r.t. June 2000 version:
– Addition of overlap capacitances CBEO and CBCO
– Change in temperature scaling of diffusion voltages
– Change in neutral base recombination current
– Addition of numerical examples with self-heating
September 2001 : Mextram level 504.1
Lower bound on Rth is now 0◦ C/W
Small changes in Fex and Q B1 B2 to enhance robustness
March 2002
iv
: Changes in implementation for increased numerical stability
Numerical stability increased of xi /Wepi at small V C1 C2
Numerical stability increased of p0∗
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Contents
August 2002— NL-UR 2002/823
Contents
Contents
v
1
Introduction
1
1.1
1
2
3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explanation of the equivalent circuit
2
2.1
General nature of the equivalent model . . . . . . . . . . . . . . . . . . .
2
2.2
Intrinsic transistor and resistances . . . . . . . . . . . . . . . . . . . . .
4
2.3
Extrinsic transistor and parasitics . . . . . . . . . . . . . . . . . . . . . .
5
2.4
Extended modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.5
Overview of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Mextram’s formulations compared to Spice-Gummel-Poon
12
3.1
General parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.2
DC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.2.1
Main current and Early effect . . . . . . . . . . . . . . . . . . .
12
3.2.2
Forward base current . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2.3
Reverse base current . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2.4
Avalanche current . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.2.5
Substrate currents . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2.6
Emitter resistance . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2.7
Base resistance . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
3.2.8
Collector resistance and epilayer model . . . . . . . . . . . . . .
19
AC model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.3.1
Overlap capacitances . . . . . . . . . . . . . . . . . . . . . . . .
19
3.3.2
Depletion capacitances . . . . . . . . . . . . . . . . . . . . . . .
20
3.3.3
Diffusion charges . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.3.4
Excess phase shift . . . . . . . . . . . . . . . . . . . . . . . . .
22
Noise model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.4.1
Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.4.2
Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.4.3
Flicker noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.3
3.4
©Koninklijke Philips Electronics N.V. 2002
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Unclassified report
3.5
Self-heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.6
Temperature model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.6.1
Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.6.2
Diffusion voltages . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.6.3
Saturation currents . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.6.4
Current gains . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.6.5
Other quantities . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
Geometric scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
3.7
4
Usage of Mextram 504
The collector epilayer model
27
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.2
Some qualitative remarks on the description of the epilayer . . . . . . . .
27
5
The substrate network
34
6
Usage of (self)-heating with Mextram
36
6.1
Dynamic heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
6.1.1
Self-heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
6.1.2
Mutual heating . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6.1.3
Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
6.1.4
Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Static heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
6.2.1
Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
6.2.2
Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
6.3
Combining static and dynamic heating . . . . . . . . . . . . . . . . . . .
39
6.4
The thermal capacitance . . . . . . . . . . . . . . . . . . . . . . . . . .
41
6.2
7
Operating point information
42
7.1
Approximate small-signal circuit . . . . . . . . . . . . . . . . . . . . . .
42
7.2
DC currents and charges . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.3
Elements of full small-signal circuit . . . . . . . . . . . . . . . . . . . .
44
References
vi
47
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
1. Introduction
August 2002— NL-UR 2002/823
1 Introduction
Mextram [1, 2] is a compact model for vertical bipolar transistors. A compact transistor
model tries to describe the I -V characteristics of a transistor in a compact way, such that
the model equations can be implemented in a circuit simulator. In principle Mextram is
the same kind of model as the well known Ebers-Moll [3] and (Spice)-Gummel-Poon [4]
models, described for instance in the texts about general semiconductor physics, Refs. [5,
6], or in the texts dedicated to compact device modelling [7, 8, 9]. These two models
are, however, not capable of describing many of the features of modern down-scaled
transistors. Therefore one has extended these models to include more effects. One of
these more extended models is Mextram, which in its earlier versions has already been
discussed in for instance Refs. [8, 10].
The complete model definition of Mextram, level 504, can be found on the web-site [1],
for instance in the report [2]. In that report a small introduction is given into the physical
basics of all the equations. An extensive physical derivation of all the model equations
is given in the report [11]. In both reports, however, the emphasis is not directly on the
designer who has to use the model. In this report, therefore, we give excerpts from the
reports above, as well as some new material, that should give a designer an idea of what
Mextram is, and how it can be used in circuit simulation. Since the bipolar transistor
model of Gummel and Poon [4] (or its Spice-implementation) is so well-known, we will
assume that the Spice-Gummel-Poon model (SGP) is known, and we will refer to it.
In Chapter 2 we will start with discussing the equivalent circuit of Mextram. This enables
us to give an overview of all the regions and the physical effects in those regions that
Mextram is able to model. It also gives an overview of the parameters of Mextram. In
Chapter 3 we will give the basic equations of Mextram and compare these, where possible
to those of Spice-Gummel-Poon (SGP). In that way a designer with knowledge of SGP
can see where the differences are, but also where many of the similarities between the
models lie. Mextram has a few main differences with SGP. The first is the more extensive
equivalent circuit, important for more accurate modelling of all kinds of RC-times. The
second is the much more accurate modelling of the collector epilayer. An introduction into
that is given in Chapter 4. Furthermore, Mextram contains a description of the parasitic
PNP, and therefore has a substrate contact. A short note about the substrate network that
should be added to Mextram is given in Chapter 5. At last, Mextram contains self-heating,
whereas SGP does not. An introduction of how to use self-heating is given in Chapter 6.
As a help to the designer most circuit simulators are capable of giving operating point
information. The kind of information that is available is discussed in Chapter 7.
1.1 Notation
To improve the clarity of the different formulas we used different typographic fonts. For
parameters we use a sans-serif font, e.g. VdE and RCv . A list of all parameters is given in
Chapter 2. For the node-voltages as given by the circuit simulator we use a calligraphic V ,
e.g. VB2 E1 and VB2 C2 . All other quantities are in normal (italic) font, like I f and q B .
©Koninklijke Philips Electronics N.V. 2002
1
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
2 Explanation of the equivalent circuit
2.1 General nature of the equivalent model
The description of a compact bipolar transistor model is based on the physics of a bipolar
transistor. An important part of this is realizing that a bipolar transistor contains various
regions, all with different doping levels. Schematically this is shown in Fig. 1. In this
report we will base our description on a NPN transistor. It might be clear that the same
model can be used for PNP transistors, using equivalent formulations. One can discern
the emitter, the base, collector and substrate regions, as well as an intrinsic part and an
extrinsic part of the transistor.
One of the steps in developing a compact model of a bipolar transistor is the creation of
an equivalent circuit. In such a circuit the different regions of the transistor are modelled
with their own elements. In Fig. 1 we have shown a simplified version of the Mextram
equivalent circuit, in which we only show the intrinsic part of the transistor, as well as
the resistances to the contacts. This simplified circuit is comparable to the Gummel-Poon
equivalent circuit.
The circuit has a number of internal nodes and some external nodes. The external nodes
are the points where the transistor is connected to the rest of the world. In our case these
are the collector node C, the base node B and the emitter node E. The substrate node S
is not shown yet since it is only connected to the intrinsic transistor via parasitics which
we will discuss later. Also five internal nodes are shown. These internal nodes are used
to define the internal state of the transistor, via the local biases. The various elements that
connect the internal and external nodes can then describe the currents and charges in the
corresponding regions. These elements are shown as resistances, capacitances, diodes and
current sources. It is, however, important to note that most of these elementary elements
are not the normal linear elements one is used to. In a compact model they describe in
general non-linear resistances, non-linear charges and non-linear current sources (diodes
are of course non-linear also). Furthermore, elements can depend on voltages on other
nodes than those to which they are connected.
For the description of all the elements we use equations. Together all these equations give
a set of non-linear equations which will be solved by the simulator. In the equations a
number of parameters are used. The equations are the same for all transistors. The value
of the parameters will depend on the specific transistor being modelled. We will give an
overview of the parameters of Mextram 504 in the next section. For a compact model it is
important that these parameters can be extracted from measurements on real transistors.
For Mextram 504 this is described in a separate report [12]. This means that the number
of parameters can not be too large. On the other hand, many parameters are needed to
describe the many different transistors in all regimes of operation.
2
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
bC
2. Explanation of the equivalent circuit
August 2002— NL-UR 2002/823
bB
bE
n + emitter
p base
RCc
&
R Bc
rB
&
1
RE
r r r rE r r
1
QE
I B1
AA
A
A
I B2
6
Æ
r r r r r r B Æ
6
Æ
Æ
r r r rC
QtE
IN
Q BE
2
I B1 B2
Q BC
Iavl
Q tC
Q epi
2
IC1 C2
n epilayer
rC
1
n + buried layer
p substrate
Figure 1: A schematic cross-section of a bipolar transistor is shown, consisting of the
emitter, base, collector and substrate. Over this cross-section we have given a simplified
equivalent circuit representation of the Mextram model, which doesn’t have the parasitics
like the parasitic PNP, the base-emitter sidewall components, and the overlap capacitances. We did show the resistances from the intrinsic transistor to the external contacts.
The current I B1 B2 describes the variable base resistance and is therefore sometimes called
R Bv . The current IC1 C2 describes the variable collector resistance (or epilayer resistance)
and is therefore sometimes called R Cv . This equivalent circuit is similar to that of the
Gummel-Poon model, although we have split the base and collector resistance into two
parts.
©Koninklijke Philips Electronics N.V. 2002
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NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
2.2 Intrinsic transistor and resistances
Let us now discuss the various elements in the simplified circuit. The precise expressions
will be given in the following chapters. Here we will only give a basic idea of the various
elements in the Mextram model. Let us start with the resistances. Node E 1 corresponds
to the emitter of the intrinsic transistor. It is connected to the external emitter node via
the emitter resistor R E . Both the collector node C and the base node B are connected to
their respective internal nodes B2 and C2 via two resistances. For the base these are the
constant resistor R Bc and the variable resistor R Bv . This latter resistor, or rather this nonlinear current source, describes DC current crowding under the emitter. Between these
two base resistors an extra internal node is present: B1 . The collector also has a constant
resistor RCc connected to the external collector node C. Furthermore, since the epilayer
is lightly doped it has its own ‘resistance’. For low currents this resistance is RCv . For
higher currents many extra effects take place in the epilayer. In Mextram the epilayer is
modelled by a controlled current source IC1 C2 .
Next we discuss the currents present in the model. First of all the main current I N gives
the basic transistor current. In Mextram the description of this current is based on the
Gummel’s charge control relation [13], see also [11]. This means that the deviations from
an ideal transistor current are given in terms of the charges in the intrinsic transistor. The
main current depends (even in the ideal case) on the voltages of the internal nodes E 1 , B2
and C2 .
Apart from the main current we also have base currents. In the forward mode these are
the ideal base current I B1 and the non-ideal base current I B2 . Since these base currents
are basically diode currents they are represented by a diode in the equivalent circuit.
In reverse mode Mextram also has an ideal and a non-ideal base current. However, these
are mainly determined by the extrinsic base-collector pn-junction. Hence they are not
included in the intrinsic transistor. The last current source in the intrinsic transistor is the
avalanche current. This current describes the generation of electrons and holes in the collector epi-layer due to impact ionisation, and is therefore proportional to the current IC1 C2 .
We only take weak avalanche into account.
At last the intrinsic transistor shows some charges. These are represented in the circuit
by capacitances. The charges Q t E and Q tC are almost ideal depletion charges resulting
from the base-emitter and base-collector pn-junctions. The extrinsic regions will have
similar depletion capacitances. The two diffusion charges Q BE and Q BC are related to
the built-up of charge in the base due to the main current: the main current consists of
mainly electrons traversing the base and hence adding to the total charge. Q BE is related
to forward operation and Q BC to reverse operation. In hard saturation both are present.
The charge Q E is related to the built-up of holes in the emitter. Its bias dependence is
similar to that of Q BE . The charge Q epi describes the built-up of charge in the collector
epilayer.
4
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
2. Explanation of the equivalent circuit
August 2002— NL-UR 2002/823
2.3 Extrinsic transistor and parasitics
After the description of the intrinsic transistor we now turn to the extrinsic PNP-region
and the parasitics. In Fig. 2 we have added some extra elements: a base-emitter side-wall
parasitic, the extrinsic base-collector regions, the substrate and the overlap capacitances.
Note that Mextram has a few flags that turn a part of the modelling on or off. In Fig. 3 on
page 7, where it is easily found for reference, the full Mextram equivalent circuit is shown
which also includes elements only present when all the extended modelling is used.
Let us start with the base-emitter sidewall parasitic. Since the pn-junction between base
and emitter is not only present in the intrinsic region below the emitter, a part of the ideal
base current will flow through the sidewall. This part is given by I BS1 . Similarly, the
sidewall has a depletion capacitance given by the charge Q tSE .
The extrinsic base-collector region has the same elements as the intrinsic transistor. We
already mentioned the base currents. For the base-collector region these are the ideal base
current Iex and the non-ideal base current I B3 . Directly connected to these currents is the
diffusion charge Q ex . The depletion capacitance between the base and the collector is
split up in three parts. We have already seen the charge Q tC of the intrinsic transistor. The
charge Q tex is the junction charge between the base and the epilayer. Mextram models
also the charge XQ tex between the outer part of the base and the collector plug.
Then we have the substrate. The collector-substrate junction has, as any pn-junction,
a depletion capacitance given by the charge Q t S . Furthermore, the base, collector and
substrate together form a parasitic PNP transistor. This transistor has a main current of
itself, given by Isub . This current runs from the base to the substrate. The reverse mode
of this parasitic transistor is not really modelled, since it is assumed that the potential of
the substrate is the lowest in the whole circuit. However, to give a signal when this is no
longer true a substrate failure current ISf is included that has no other function than to
warn a designer that the substrate is at a wrong potential.
Finally the overlap capacitances CBEO and CBCO are shown that model the constant capacitances between base and emitter or base and collector, due to for instance overlapping
metal layers (this should not include the interconnect capacitances).
2.4 Extended modelling
In Mextram two flags can introduce extra elements in the equivalent circuit (see Fig. 3 on
page 7). When EXPHI = 1 the charge due to AC-current crowding in the pinched base
(i.e. under the emitter) is modelled with Q B1 B2 . (Also another non-quasi-static effect,
base-charge partitioning, is then modelled.) When EXMOD = 1 the external region is
modelled in some more detail (at the cost of some loss in the convergence properties
in a circuit simulator). The currents Iex and Isub and the charge Q ex are split in two
parts, similar to the splitting of Q tex . The newly introduced elements are parallel to the
charge Q tex .
©Koninklijke Philips Electronics N.V. 2002
5
NL-UR 2002/823— August 2002
brC
CBCO
Usage of Mextram 504
brB
Unclassified report
brE
CBEO
n + emitter
p base
r
Q tSE
r r r r rE r r
HH
I BS
&
1
QE
1
r
r r r
Æ
Æ?AA
r r r r rC
6
Æ
Æ
r r r
r
RCc
&
R Bc
B1
Isub
bS
6
Æ
r r r r r r B Æ
6
Æ
Æ
r r r rC
QtE
IN
Q BC
Iavl
Q tC
Q epi
Q ex
Q tS
AA
A
A
I B2
2
Q tex
Iex +I B3
ISf
I B1
Q BE
I B1 B2
XQ tex
RE
1
n epilayer
n + buried layer
p substrate
Figure 2: Shown is the Mextram equivalent circuit for the vertical NPN transistor, without
extra modelling (i.e. EXMOD = 0 and EXPHI = 0). As in Fig. 1 we have schematically
shown the different regions of the physical transistor.
6
©Koninklijke Philips Electronics N.V. 2002
2
IC1 C2
Unclassified report
brC
2. Explanation of the equivalent circuit
brB
CBCO
August 2002— NL-UR 2002/823
brE
CBEO
n + emitter
p base
r
Q tSE
r r r r rE r r
HH
I BS
1
&
QE
1
r
rB r r r r
r r r
AA Æ
Æ? Æ
Æ?AA
r r r
r r r r
C
6
Æ
Æ
bS
r r r r
&
XQ tex
R Bc
1
Isub
Q tex
Iex +I B3
XQ ex
Q tS
QtE
IN
2
Q BC
Iavl
Q tC
Q epi
Q ex
ISf
AA
A
A
Q BE
I B1 B2
XIex XIsub
I B1
I B2
6
Æ
r r r r r r r B Æ
6
Æ
Æ
r r r rC
Q B1 B2
RCc
RE
1
2
IC1 C2
n epilayer
n + buried layer
p substrate
Figure 3: The full Mextram equivalent circuit for the vertical NPN transistor. Schematically the different regions of the physical transistor are shown. The current I B1 B2 describes the variable base resistance and is therefore sometimes called R Bv . The current
IC1 C2 describes the variable collector resistance (or epilayer resistance) and is therefore
sometimes called RCv . The extra circuit for self-heating is discussed in Chapter 6.
©Koninklijke Philips Electronics N.V. 2002
7
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
2.5 Overview of parameters
In this section we will give an overview of the parameters used in Mextram. These parameters can be divided into different categories. In the formal description these parameters
are given by a letter combination, e.g. IS. In equations however we will use a different
notation for clarity, e.g. Is . Note that we used a sans-serif font for this. Using this notation
it is always clear in an equation which quantities are parameters, and which are not. Many
of the parameters are dependent on temperature. For this dependence the model contains
som extra parameters. When the parameter is corrected for temperature it is denoted by an
index T, e.g. IsT . In the formal documentation [1, 2] the difference between the parameter
at reference temperature Is and the parameter after temperature scaling IsT is made in a
very stringent way. In this report however we don’t add the temperature subscript, unless
it is needed.
First of all we have some general parameters. Flags are either 0 when the extra modelling
is not used, or 1 when it is.
LEVEL
EXMOD
EXPHI
EXAVL
MULT
LEVEL Model level, here always 504
EXMOD Flag for EXtended MODelling of the external regions
EXPHI
Flag for extended modelling of distributed HF effects in
transients
EXAVL Flag for EXtended modelling of AVaLanche currents
MULT
Number of parallel transistors modelled together
As mentioned in the description of the equivalent circuit some currents and charges are
split, e.g. in an intrinsic part and an extrinsic part. Such a splitting needs a parameter.
There are 2 for the side-wall of the base-emitter junction. Then the collector-base region
is split into 3 parts, using 2 parameters.
XIB1
XCjE
XIBI
XCJE
XCjC
XCJC
Xext
XEXT
Fraction of the ideal base current that goes through the sidewall
Fraction of the emitter-base depletion capacitance that belongs
to the sidewall
Fraction of the collector-base depletion capacitance that is under
the emitter
Fraction of external charges/currents between B and C1 instead
of B1 and C1
A transistor model must in the first place describe the currents, and we use some parameters for this. The main currents of both the intrinsic and parasitic transistors are described
by a saturation current and a high-injection knee current. We also have two Early voltages for the Early effect in the intrinsic transistor. The two ideal base currents are related
to the main currents by a current gain factor. The non-ideal base currents are described
by a saturation current and non-ideality factor or a cross-over voltage (due to a kind of
high-injection effect). The avalanche current is described by three parameters.
8
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Is
Ik
ISs
Iks
Vef
Ver
βf
βri
IBf
mLf
IBr
VLr
Wavl
Vavl
Sfh
2. Explanation of the equivalent circuit
IS
IK
ISS
IKS
VEF
VER
BF
BRI
IBF
MLF
IBR
VLR
WAVL
VAVL
SFH
August 2002— NL-UR 2002/823
Saturation current for intrinsic transistor
High-injection knee current for intrinsic transistor
Saturation current for parasitic PNP transistor
High-injection knee current for parasitic PNP transistor
Forward Early voltage of the intrinsic transistor
Reverse Early voltage of the intrinsic transistor
Current gain of ideal forward base current
Current gain of ideal reverse base current
Saturation current of the non-ideal forward base current
Non-ideality factor of the non-ideal forward base current
Saturation current of the non-ideal reverse base current
Cross-over voltage of the non-ideal reverse base current
Effective width of the epilayer for the avalanche current
Voltage describing the curvature of the avalanche current
Spreading factor for the avalanche current
Mextram contains both constant and variable resistances. For variable resistances the
resistance for low currents is used as a parameter. The epilayer resistance has two extra
parameters related to velocity saturation and one smoothing parameter.
RE
RBc
RBv
RE
RBC
RBV
RCc
RCv
SCRCv
Ihc
axi
RCC
RCV
SCRCV
IHC
AXI
Constant resistance at the external emitter
Constant resistance at the external base
Low current resistance of the pinched base (i.e. under the
emitter)
Constant resistance at the external collector
Low current resistance of the epilayer
Space charge resistance of the epilayer
Critical current for hot carriers in the epilayer
Smoothing parameter for the epilayer model
All depletion capacitances are given in terms of the capacitance at zero bias, a built-in or
diffusion voltage and a grading coefficient (typically between the theoretical values 1/2
for an abrupt junction and 1/3 for a graded junction). The collector depletion capacitance is limited by the width of the epilayer region. Its intrinsic part also has a current
modulation parameter.
Cj E
pE
VdE
Cj C
pC
VdC
Xp
CJE
PE
VDE
CJC
PC
VDC
XP
Depletion capacitance at zero bias for emitter-base junction
Grading coefficient of the emitter-base depletion capacitance
Built-in diffusion voltage emitter-base
Depletion capacitance at zero bias for collector-base junction
Grading coefficient of the collector-base depletion capacitance
Built-in diffusion voltage collector-base
Fraction of the collector-base depletion capacitance that is
constant
©Koninklijke Philips Electronics N.V. 2002
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NL-UR 2002/823— August 2002
mC
Cj S
MC
CJS
pS
PS
VdS
VDS
Usage of Mextram 504
Unclassified report
Current modulation factor for the collector depletion charge
Depletion capacitance at zero bias for collector-substrate
junction
Grading coefficient of the collector-substrate depletion
capacitance
Built-in diffusion voltage collector-substrate
New in Mextram 504 are two constant overlap capacitances.
CBEO
CBCO
CBEO
CBCO
Base-emitter overlap capacitance
Base-collector overlap capacitance
Most of the diffusion charges can be given in terms of the DC parameters. For accurate AC-modelling, however, it is better that DC effects and AC effects have their own
parameters, which in this case are transit time parameters.
τE
mτ
τB
τepi
τR
TAUE
MTAU
TAUB
TEPI
TAUR
(Minimum) transit time of the emitter charge
Non-ideality factor of the emitter charge
Transit time of the base
Transit time of the collector epilayer
Reverse transit time
Noise in the transistor is modelled by using only three extra parameters.
Kf
KfN
Af
KF
KFN
AF
Flicker-noise coefficient of the ideal base current
Flicker-noise coefficient of the non-ideal base current
Flicker-noise exponent
Then we have the temperature parameters. First of all two parameters describe the temperature itself. Next we have some temperature coefficients, that are related to the mobility
exponents in the various regions. We also need some bandgap voltages to describe the
temperature dependence of some parameters.
Tref
dTa
TREF
DTA
AQB0
AE
AB
Aepi
Aex
AC
AS
AQBO
AE
AB
AEPI
AEX
AC
AS
10
Reference temperature
Difference between local ambient and global ambient
temperatures
Temperature coefficient of zero bias base charge
Temperature coefficient of RE
Temperature coefficient of RBv
Temperature coefficient of RCv
Temperature coefficient of RBc
Temperature coefficient of RCc
Temperature coefficient of the mobility related to the substrate
currents
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
dVgββ f
dVgββ r
VgB
VgC
VgS
Vgj
dVgττE
2. Explanation of the equivalent circuit
DVGBF
DVGBR
VGB
VGC
VGS
VGJ
DVGTE
August 2002— NL-UR 2002/823
Difference in bandgap voltage for forward current gain
Difference in bandgap voltage for reverse current gain
Bandgap voltage of the base
Bandgap voltage of the collector
Bandgap voltage of the substrate
Bandgap voltage of base-emitter junction recombination
Difference in bandgap voltage for emitter charge
New in Mextram 504 are two formulations that are dedicated to SiGe modelling. For a
graded Ge content we have a bandgap difference. For recombination in the base we have
another parameter.
dEg
Xrec
DEG
XREC
Bandgap difference over the base
Pre-factor for the amount of base recombination
Also new in Mextram 504 is the description of self-heating, for which we have the two
standard parameters.
Rth
Cth
RTH
CTH
Thermal resistance
Thermal capacitance
©Koninklijke Philips Electronics N.V. 2002
11
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Usage of Mextram 504
Unclassified report
3 Mextram’s formulations compared to Spice-GummelPoon
In this chapter we will give a summary of the most important equations of Mextram. We
will compare them, where possible, with those of Spice-Gummel-Poon (SGP), and note
the differences and similarities:
Mextram formulation;
SGP formulation.
In all the equations we will use the Mextram formulation on the left and the equivalent
SGP formulation on the right. We will also give the equivalence between parameters if
present. For clarity, our comparison is based on the Pstar SGP implementation (known
as bipolar transistor model, level 2). Sometimes the same SGP parameter is known under
different names. We will indicate this, for example, as Tref /Tamb .
3.1 General parameters
Let us start with some general parameters.
Mextram SGP
comments
LEVEL LEVEL Typical simulator dependent parameter.
MULT
AREA/M Naming depends on simulator. In most if not all cases noninteger values are allowed.
Tref
Tref /Tamb The reference temperature might vary in naming, but always has
the same function.
DTA
DTA
The increase in local ambient temperature is not present in all
simulators. See also Chapter 6.
EXMOD —
In Mextram three parameters are used as explicit flags. In SGP
only the electrical parameters can act as a flag, like IRB .
EXPHI
—
EXAVL —
3.2 DC model
3.2.1 Main current and Early effect
The forward part of the main current is in both models almost equal:
I f = Is eVB2 E1 /VT ;
I f = Is eVB1 E1 /NF VT .
(3.1)
The main difference is the non-ideality in the SGP formulation. The non-ideality factor
or emission coefficient NF is not present in Mextram. Experience has learned that the
emission coefficient for Si-based processes is not needed. Any non-ideality in the collector current is due to the reverse-Early effect and will be modelled via the normalised base
charge.
12
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Mextram’s formulations compared to SGP
August 2002— NL-UR 2002/823
IC (µA)
1.3
1.2
1.1
0
2
4
6
8
V CB (V)
10
12
Figure 4: Collector current as function of collector-base bias. The markers are from
measurements. The solid line is the result of Mextram. The curvature in the current
can clearly be observed. This curvature also means that the output conductance is not
constant, as in SGP. The dashed lines is also from Mextram, but without avalanche.
For the reverse main currents the same holds
Ir = Is eVB2 C2 /VT ;
Ir = Is eVB1 C1 /NR VT .
(3.2)
Again the non-ideality factor is not present in Mextram.
The main current is now, in both cases
IN =
I f − Ir
;
qB
I1 − I2 =
I f − Ir
.
qB
(3.3)
The normalised base charge is something like
qB =
Q B0 + Q t E + Q tC + Q B E + Q BC
,
Q B0
(3.4)
where Q B0 is the base charge at zero bias. The total charge consists of Q B0 , the baseemitter and base-collector depletion charges and the base-emitter and base-collector diffusion charges. The normalised base charge can be given as a product of the Early effect
(describing the variation of the base width given by the depletion charges) and a term
which includes high injection effects. The Early effect term is in both models given using
the normalised base charge q1 = 1+ Q t E /Q B0 + Q tC /Q B0 , but in a different formulation:
Vt
Vt
q1 = 1 + E + C ;
Ver
Vef
©Koninklijke Philips Electronics N.V. 2002
q1 = 1 −
VB1 E1
Var
−
VB1 C1
Vaf
−1
.
(3.5)
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NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
100
VForw−Early(V)
90
80
70
60
50
0
2
4
6
V CB (V)
8
10
12
Figure 5: The actual forward Early voltage obtained by numerical differentiation of the
collector current in Fig. 4: VForw−Early = IC /(dIC /dVCB ). SGP would give a constant
value of this Early voltage. Again the dashed line is the Mextram result without avalanche.
The voltages Vt E and VtC describe the curvature of the depletion charges as function of
junction biases, but not their magnitude: Q t E = CjE · Vt E and Q tC = CjC · VtC (taking only
the intrinsic parts into account). The difference in the Early effect modelling between SGP
and Mextram is that Mextram includes the bias-dependence of the Early effect. Using the
normalised depletion charges the actual Early voltage can be a function of current. Hence
the collector current will not really be a straight line as function of collector bias, but it
will show some curvature. This has been shown in Figs. 4 and 5. In SGP the Early effect
is modelled by linearising the effect around zero bias.
Note that the Early voltage parameter here is only V ef = 44 V. So the parameter is much
smaller than the actual Early voltage!
Mextram is capable of modelling the effect of a gradient in the Ge-content in the base,
as an extra option using the parameter dEg [11, 14]. This will change the Early effects.
This can mainly be seen in the sharp decrease of the current gain even before high-current
effects (Webster effect, Kirk effect) start to play a role. We will not further discuss this
here.
The influence of the diffusion charges on the current (Webster effect, giving a knee in the
current) is modelled in the normalised base charge q B as
√
1 + 4q2 + 1
1
1
,
(3.6)
q B = q1 (1 + 2 n 0 + 2 n B );
q B = q1
2
n0 =
14
4I f /Ik
p
;
1 + 1 + 4I f /Ik
q2 =
If
Ir
+
,
IKF
IKR
(3.7)
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
nB =
Mextram’s formulations compared to SGP
August 2002— NL-UR 2002/823
4Ir /Ik
p
.
1 + 1 + 4Ir /Ik
(3.8)
Although the formulations seem to be very different, the effect of both formulations is
nearly the same. If one takes for instance Ir = 0 the results match. So the only difference
is in the situation where I f and Ir are both important. In SGP this is in hard-saturation. In
Mextram the same happens in quasi-saturation. Since the latter regime is quite important
in modern bipolar processes, Mextram gives a better result here.
Is
—
—
Vef
Is
NF
NR
Vaf
Ver
Var
dEg
—
Ik
IKF & IKR
Almost the same
Not needed in Mextram
Not needed in Mextram
The forward Early voltages are present in both models. The
Mextram value is, however, not close to the actual Early voltage,
due to the bias-dependence. The Mextram value can be a factor
2 smaller.
The reverse Early voltages are present in both models. The
Mextram value is, however, not close to the actual Early voltage, due to the bias-dependence.
Including a gradient in the Ge-content of the base is an extra
option of Mextram.
Mextram contains only one knee current.
3.2.2 Forward base current
The forward base currents consist of an ideal part and a non-ideal part. Both parts are
very similar to each other in the two models. For the ideal part we have
Is VB E /VT
Is VB E /VT
e 2 1
e 1 1
−1 ;
IR E =
−1 .
(3.9)
I B1 =
βf
βf
In Mextram the ideal forward base current can be split into a bottom and a sidewall component using the parameter XIB1 . Mextram has an option to include the collector-voltage
dependence of this base current, like in the case of neutral base recombination, using the
parameter Xrec [11, 14].
Note that due to the reverse Early effect, shown for instance in Fig. 6, the actual current
gain might be much smaller than the parameter value βf ! The current gain parameter βf
is the limit of the current gain for VBE to zero, if there were no non-ideal base current. For
the data of Fig. 6 we have βf ' 400.
Also the non-ideal base current is almost the same in the two models
I L E = ISE eVB1 E1 /NE VT − 1 .
I B2 = IBf eVB2 E1 /mLf VT − 1 ;
©Koninklijke Philips Electronics N.V. 2002
(3.10)
15
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
200
hfe [–]
150
100
50
0
0.2
0.4
0.6
0.8
VBE [V]
1.0
1.2
Figure 6: The current gain as function of base-emitter voltage for a SiGe process which
has a large reverse-Early effect (an effectively small reverse Early voltage). From
Ref. [14].
βf
XIB1
Xrec
IBf
mLf
βf
—
—
ISE
NE
Almost the same
No extra splitting of the base current is available in SGP.
Neutral base recombination is an extra option of Mextram.
The same
The same
3.2.3 Reverse base current
The reverse base currents are very similar to the forward base currents. For the ideal part
in Mextram, however, the knee due to electrons injected in the base is the same as that of
the main current. Note that in Mextram the reverse base current only exists in the extrinsic
regions
Iex
Is
=
βri
2 eVB1 C1 /VT − 1
1+
q
1 + Is eVB1 C1 /VT /Ik
;
I RC
Is VB C /VT
1
1
e
=
−1 .
βr
(3.11)
In Mextram the ideal reverse base current can be split into two extrinsic base currents,
using Xext . The same is done with the extrinsic capacitance, as we discuss later.
The non-ideal base current is somewhat different in both models. The SGP model contains
a standard description of a non-ideal diode. Mextram contains a description based on SRH
recombination where for small biases again a ideal slope exists. The cross-over from ideal
16
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Mextram’s formulations compared to SGP
August 2002— NL-UR 2002/823
slope to non-ideal slope is determined by the cross-over voltage VLr .
eVB1 C1 /VT − 1
I B3 = IBr V /2V
;
e B1 C1 T + eVLr /2VT
I LC = ISC eVB1 C1 /NC VT − 1 .
(3.12)
The two equations become the same in the very normal case of a low cross-over voltage
VLr and a non-ideality factor of NC = 2.
βri
βr
IBr
VLr
—
Xext
ISE
—
NE
—
Almost the same. Mextram contains a knee in the reverse base
current. Furthermore, in Mextram βri only describes the intrinsic current gain, so not including substrate current effects.
The same
Mextram has a cross-over voltage,
where SGP has a non-ideality factor
In Mextram splitting the extrinsic currents and charges is
possible.
3.2.4 Avalanche current
Since SGP does not contain an avalanche model, we can not compare the models. In
Mextram the avalanche current is given as
Iavl = IC1 C2 × G(VB1 C1 , IC1 C2 )
(3.13)
where the generation factor, related to the multiplication factor G = M − 1, is a function
of bias and current. It is supposed to be small (G 1). For some results, see Figs. 4
and 5. Mextram is capable of modelling snapback effects at high currents, but only as an
option.
In some SGP implementations an empirical model is present along the lines of



Iavl = IC1 C2 
1−
1
V
C1 B1

N − 1 . (3.14)
BVCBO
Real breakdown is not modelled in Mextram. One of the most important reasons being
that breakdown effects are really bad for convergence.
Wavl
Vavl
Sfh
—
—
—
No avalanche model present in SGP
No avalanche model present in SGP
No avalanche model present in SGP
©Koninklijke Philips Electronics N.V. 2002
17
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
3.2.5 Substrate currents
Mextram contains a substrate current, simply modelled as
V
B1 C1 /VT
2 ISs e
−1
q
Isub =
.
V
B1 C1 /VT
1 + 1 + Is e
/Iks
(3.15)
This substrate current describes the main current of the parasitic PNP, i.e. it describes the
holes going from the base to the substrate. The current that runs in the case of a forward
bias on the substrate-collector junction is not modelled in a physical way, since this should
not happen anytime. There is only a signal current I S f to alert a designer to this wrong
bias situation.
ISs
Iks
—
—
SGP has no substrate current
SGP has no substrate current
3.2.6 Emitter resistance
The emitter resistance is in both models constant: RE .
RE
RE
The same
3.2.7 Base resistance
The base resistance consists of a constant extrinsic part and a variable intrinsic part. Both
Mextram and SGP contain both parts. In SGP, however, the two are lumped together into
one resistance in the equivalent circuit.
The variable resistance in SGP can be due either to conductivity modulation (i.e. more
charge in the base, modelled by q B ) or due to current crowding (modelled using the parameter IRB ). In Mextram both effects are taken into account simultaneously. The current
crowding effect is based on the same theory as that of SGP [15, 16].
SGP models the variable part of the resistance as function of the current through it. This
is done also in many publications about the subject. The problem then is that one has to
decide whether the DC resistance or the small-signal resistance is taken. In Mextram the
current through the resistance is modelled as function of the applied voltage. In that way
both the DC resistance as well as the small-signal resistance follow directly form either
I / V or dI /dV .
For the variable part we have in Mextram
I B1 B2 =
18
qB
[2VT (eVB1 B2 /VT −1)+VB1B2 ].
3 RBv
(3.16)
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Mextram’s formulations compared to SGP
August 2002— NL-UR 2002/823
It is important to realise that the actual intrinsic base resistance can be much smaller
than RBv , due to current crowding, or, more likely, due to the extra charge in the base
modelled by q B !
For SGP we have either
I B B1 =
qB
VBB1 ,
RB − RBM
(3.17)
I B B1 =
f (I B B1 , IRB )
VBB1 .
3 (RB − RBM )
(3.18)
or
RBc
RBM
RBv
RB −RBM
—
IRB
The extrinsic part of the base resistance is the same in both models: constant.
Mextram has as parameter the zero-bias value of the intrinsic
part. SGP has the zero-bias value of the total base resistance.
Empirical parameter of SGP for a description of current crowding. Not needed in Mextram.
3.2.8 Collector resistance and epilayer model
The collector resistance not including the collector epilayer is in both models constant.
SGP does not have a collector epilayer model for the current. (For the charges an empirical model is present.) It is this epilayer model which is maybe the largest improvement
of Mextram over SGP. For this reason we discuss it in more detail in Chapter 4.
RCc
RCv
SCRCv
Ihc
axi
RC
—
—
—
—
The same
SGP has no collector epilayer model
SGP has no collector epilayer model
SGP has no collector epilayer model
SGP has no collector epilayer model
3.3 AC model
3.3.1 Overlap capacitances
SGP has no overlap capacitances. Mextram has constant overlap capacitances. These
are meant for modelling all kinds of constant parasitic capacitances that belong to the
transistor. Most of them are vertical capacitances, say between poly layers or over STI,
but also the effect of contacts is included. All interconnect capacitances are not included.
©Koninklijke Philips Electronics N.V. 2002
19
NL-UR 2002/823— August 2002
CBEO
CBCO
—
—
Usage of Mextram 504
Unclassified report
SGP has no overlap capacitances.
SGP has no overlap capacitances.
3.3.2 Depletion capacitances
The depletion capacitances of Mextram are not too different from those of SGP. The basic
formulation is
C=
Cj
.
(1 − V /Vd )p
(3.19)
Note that the notation of the parameters differs. Furthermore, in both models not the
capacitance, but the charge is implemented.
There are a few differences, however.
• Mextram does not linearise the capacitances above a certain forward bias. It uses a
smooth transition to a constant capacitance, for which no parameter is available. Since
the transition in both models happens in the region where diffusion capacitances are dominant the difference in capacitance is not very important. For higher order derivatives,
however, the SGP will give discontinuities.
• Mextram has a split in the base-emitter capacitance and an extra split in the basecollector capacitance. Having the extra nodes and having these splits in the capacitances
gives a more accurate RC network. This means a better description of RC-times and
hence of the small-signal parameters (e.g. Y -parameters).
• Reach-through of the base-collector capacitance is modelled, such that the capacitance
becomes constant for large reverse biases. The basic formulation is
C=
(1 − Xp )CjC
p + Xp CjC .
1 − V /VdC C
(3.20)
• The base-collector capacitance is current dependent. This is related to the finite velocity
of the electrons in the depletion layer, see Chapter 4.
Cj E
VdE
pE
Cj C
VdC
pC
Cj S
VdS
pS
XCjC
20
Cj E
VjE /PE
ME
Cj C
VjC /PC
MC
Cj S
VjS /PS
MS
XCjC
The same
The same, apart from notation
The same, apart from notation
The same
The same, apart from notation
The same, apart from notation
The same
The same, apart from notation
The same, apart from notation
The same
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
XCjE
Xext
XCjE
Xext
Xp
mC
—
—
—
FC
Mextram’s formulations compared to SGP
August 2002— NL-UR 2002/823
SGP does not contain a split in the base-emitter capacitance
SGP does not contain an extra split of the base-collector
capacitance
SGP does not model reach-through.
Mextram has a current dependence in the base-collector
capacitance.
Mextram does not linearise the capacitances. Instead it the capacitances level off smoothly. For this model-constants a j E , a jC ,
and a j S are used.
3.3.3 Diffusion charges
For the low-current transit time there is not a large difference between Mextram and SGP.
Only, Mextram contains two contributions, one from the emitter charge and one from the
base charge. The contribution from the base charge is
Q B E = q1τB I f
1+
p
2
1 + 4I f /Ik
;
QED =
1
2
p
τFF I f
. (3.21)
q1
1 + 1 + 4I f /Ikf
In the SGP formulation we neglected here the contribution of Ir , which is zero anyway
in normal operation, even in quasi-saturation/Kirk effect. Note that also the Early effect
(via q1 ) is taken into account differently. In Mextram there is a second contribution from
the charged stored in the emitter (or the neutral part of the base-emitter depletion layer),
which does not have a knee, but does have a non-ideality factor
Q E = τE Is eVB2 E1 /mτ VT .
(3.22)
For higher currents the models start to differ quite a lot. SGP models the increase in
effective transit time due to base-widening/quasi-saturation/Kirk effect as
"
#
2
If
eVB1 C1 /1.44VTF .
τFF = τF 1+XTF
I f +ITF
(3.23)
In Mextram there is a collector epilayer model with a finite voltage drop. As a result the
bias over the intrinsic base-collector junction, VB2 C2 , can become forward biased, such
that the reverse part of the main current, Ir , becomes comparable to the forward part I f .
This results in an extra charge in the base
Q BC = q1τB Ir
1+
p
2
1 + 4Ir /Ik
.
(3.24)
Furthermore, there will be a lot of charge in the collector epilayer, due to the basewidening
xi 2
Q epi ' τepi
Iepi .
(3.25)
Wepi
©Koninklijke Philips Electronics N.V. 2002
21
NL-UR 2002/823— August 2002
Usage of Mextram 504
Unclassified report
where xi /Wepi is the size of the base widening or thickness of the injection region, compared to the epilayer thickness, and Iepi is the current through the epilayer, normally
equal to the main collector current I N . In hard saturation when the current vanishes but
the base-collector junction is still in forward, the real expression for Q epi is such that it
still describes the large built-up of charge in the epilayer.
τE +ττB
τF
mτ
τepi
—
—
—
XTF
—
ITF & VTF
τR
τR
The low-current transit time in Mextram contains both a contribution from the emitter diffusion charge, as well as from the
base-diffusion charge.
SGP does not have an emitter diffusion charge.
A parameter which belongs directly to the quasi-saturation
charge in the epilayer is not present in SGP. The effect is modelled in another way.
The effect of increasing transit time due to quasi-saturation is
modelled in Mextram via the collector epilayer model. This
increase depends very much on τepi , but also on a lot of other
parameters. Maybe τF · XTF could be compared to τepi , in the
kind of effect they have.
See XTF . In practice ITF should be of the same order as Ihc , and
VTF should be of the same order as RCv Ihc or SCRCv Ihc .
The reverse transit times are almost equivalent. But note that in
Mextram it is used only in the extrinsic regions.
3.3.4 Excess phase shift
Both SGP and Mextram model excess phase shift. The difference in modelling is, however, quite different. The first thing to realise is that the most important contribution to
the excess phase shift does not come from the intrinsic model. Instead the extrinsic capacitances and resistances (mainly the base resistance, the base-collector capacitance and
its split using the parameter Xext ) are much more important. Only when these are correct, it is useful to look at non-quasi-static effects in the intrinsic transistor that can also
cause excess phase shift. Since Mextram has a much better equivalent circuit, the intrinsic
excess phase shift gives only a marginal effect.
In Mextram the intrinsic excess phase shift is modelled using base-charge partitioning
(and only when EXPHI = 1). This means that a part of the charge Q B E , which is VB2 E1 driven, is not supplied totally by the emitter, but also partly by the collector. To this end
the charges Q B E and Q BC are changed from their previously calculated values to
Q BC →
QBE →
1
3
2
3
Q B E + Q BC ,
(3.26)
QBE.
(3.27)
In SGP excess phase shift is implemented using a Bessel approximation. Calculations are
22
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Mextram’s formulations compared to SGP
August 2002— NL-UR 2002/823
not done with I1 = I f /q B , but with I X , which is a solution of the differential equation
3ω02 I1 =
d2 I X
dIx
+ 3ω02 I X , (3.28)
+ 3ω0
2
dt
dt
where ω0 = 1/(TFF · PTF π/180).
Mextram also models an extra effect in the lateral direction (AC current crowding). Again
only when EXPHI = 1 a charge is added parallel to the intrinsic base resistance
(3.29)
Q B1 B2 = 15 VB1 B2 Ct E + C B E + C E .
—
PTF
Excess phase shift within Mextram has no extra parameter (besides, say, the base resistances and Xext ), but Mextram does
have a switch (EXPHI) to turn the model on or off.
3.4 Noise model
The noise models of Mextram and SGP do not differ very much. But Mextram has a larger
equivalent circuit, and hence more currents/resistances and corresponding noise sources.
3.4.1 Thermal noise
All resistances have thermal noise, including the variable part of the base resistance. For
the variable collector resistance, the epilayer model of Mextram, a more advanced model
is used for the case of base-widening.
3.4.2 Shot noise
All diode-like currents have shot-noise. This includes the main current, the main substrate
current (not present in SGP) and all (forward and reverse) base currents.
3.4.3 Flicker noise
All base currents also have flicker noise (1/ f -noise), modelled with a pre-factor Kf and a
power Af . In Mextram the non-ideal forward base current has its own pre-factor KfN and
a fixed power 2.
Af
Kf
KfN
Af
Kf
—
The same
The same
Not present in SGP
©Koninklijke Philips Electronics N.V. 2002
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Usage of Mextram 504
Unclassified report
3.5 Self-heating
SGP does not contain self-heating. Mextram does. See Chapter 6.
Rth
Cth
—
—
SGP does not contain self-heating
SGP does not contain self-heating
3.6 Temperature model
SGP has only 3 temperature parameters, whereas Mextram has 14 in total. This means
that various Mextram parameters will correspond to the same SGP parameter. For the
temperature scaling, we first need some definitions:
TK = TEMP + DTA + 273.15 + VdT ,
TR K = Tref + 273.15,
TK
tN =
,
T
R K
k
TK ,
VT =
q
k
TR K ,
VTR =
q
1
1
1
=
−
.
V1T
VT
VTR
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
Here TK is the actual temperature including self-heating (see Chapter 6).
3.6.1 Resistances
In SGP the resistances are constant over temperature. In Mextram they all have their own
parameter that is linked to the temperature dependence of the mobility.
A
RET = RE t N E ,
AB −AQB0
RBv T = RBv t N
RBcT = RBc t NAex ,
A
RCcT = RCc t N C ,
A
RCv T = RCv t N epi .
(3.36)
,
(3.37)
(3.38)
(3.39)
(3.40)
The parameter AQB0 is not related to a mobility, but is an effective parameter describing
the temperature scaling of the zero-bias base charge.
24
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
AE
AB
Aex
AC
Aepi
AQB0
Mextram’s formulations compared to SGP
—
—
—
—
—
—
August 2002— NL-UR 2002/823
SGP has no temperature scaling for resistances
SGP has no temperature scaling for resistances
SGP has no temperature scaling for resistances
SGP has no temperature scaling for resistances
SGP has no temperature scaling for resistances
SGP has no temperature scaling for zero bias base charge
3.6.2 Diffusion voltages
The equation for the diffusion voltages is basically the same.
VdT = −3VT ln t N + Vd t N + (1 − t N )Vg .
(3.41)
Only Mextram makes sure that they do not become negative. The important parameter
for the temperature scaling of the diffusion voltages is the bandgap Vg . SGP has only one
bandgap: Eg .
Once the temperature scaling of the diffusion voltages has been done, the depletion capacitances in Mextram scale basically as
Vd p
CjT = Cj
.
(3.42)
VdT
where the grading coefficient p is used.
VgB
VgC
VgS
Eg
Eg
Eg
Used for base-emitter diffusion voltage
Used for base-collector diffusion voltage
Used for substrate-collector diffusion voltage
3.6.3 Saturation currents
The equation for the saturation current is also similar
4−AB −AQB0
IsT = Is t N
e−VgB /V1T ;
IsT = Is t N ti e−Eg /V1T .
X
(3.43)
The same bandgap as before is used. The power of the pre-factor is different.
For the saturation current of the forward non-ideal base current we have
(6−2mLf )
IBf T = IBf t N
e−VgJ /mLf
V1T
;
X −Xtb
ISET = ISE t N ti
e−Eg /NE
V1T
.
(3.44)
V1T
.
(3.45)
For the saturation current of the reverse non-ideal base current we have
IBr T = IBr t N2 e−VgC /2 V1T ;
X −Xtb
ISCT = ISC t N ti
e−Eg /NC
The saturation current of the substrate scales with temperature as
4−AS
ISsT = ISs t N
e−VgS /V1T .
©Koninklijke Philips Electronics N.V. 2002
(3.46)
25
NL-UR 2002/823— August 2002
VgB
VgJ
VgC
(AB )
Eg
Eg
Eg
Xti
VgS
AS
—
—
Usage of Mextram 504
Unclassified report
Used for collector saturation current
Used for forward non-ideal base current
Used for reverse non-ideal base current
AB and Xti have kind of the same purpose in the temperature
scaling of the collector saturation current
SGP has no substrate current
SGP has no substrate current
3.6.4 Current gains
The temperature modelling of the current gains is different. In SGP a simple power dependence is used. In Mextram the model uses the normally small difference in bandgap
between either emitter and base or base and collector. This can be important in the case
of SiGe transistors, where the bandgap difference can become larger.
AE −AB −AQB0
βf T = βf t N
e−dVgββ f /V1T ; βf T = βf t N tb ,
βriT = βri e−dVgββ r /V1T ;
dVgββ f
dVgββ r
—
—
—
Xtb
X
X
βr T = βr t N tb .
(3.47)
(3.48)
SGP uses Xtb
SGP uses Xtb
Mextram uses dVgββ f and dVgββ r
3.6.5 Other quantities
Mextram contains also temperature scaling of the Early voltages, the transit times and
the material constant for the avalanche model. Since SGP does not have any equivalents,
these are not discussed here.
dVgττ E
—
Specially for the emitter transit time
3.7 Geometric scaling
Geometric scaling is not present in both models. In both cases, geometric scaling can and
should be done in a pre-processing way. There are no fundamental differences between
Mextram and SGP in this respect. The only difference is that Mextram has a more complete equivalent circuit and is therefore capable of somewhat more advanced geometric
scaling [12].
26
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
4. The collector epilayer model
August 2002— NL-UR 2002/823
4 The collector epilayer model
4.1 Introduction
The collector epilayer of a bipolar transistor is the most difficult part to model. The reason
for this is that a number of effects play a role and act together. Since in Spice-GummelPoon a model for this epilayer is not present at all, this chapter is devoted to a short
introduction of the physics that plays a role. We will restrict ourselves to modelling the
epilayer in as far as it is part of the intrinsic transistor. The current Iepi = IC1 C2 through the
epilayer is for low current densities mainly determined by the main current I N . Hence the
epilayer is a part of the transistor that is current driven. Since the dope concentration in the
epilayer is in general small, high injection effects are important. In that case the epilayer
will be (partly) flooded by holes and electrons. Even though then the main current and
the epilayer current depend on each other and their equations become coupled, we will
still consider the epilayer to be current driven, i.e. our model will have Iepi as a starting
quantity.
The regions where no injection takes place can either be ohmic, which implies charge
neutral, or depleted. In depletion regions the electric field is large and the electrons will
therefore move with the saturation velocity. These electrons can be called hot carriers.
The electrons will contribute to the charge. In case of large currents this moving charge
becomes comparable to the dope, the net charge decreases, or even changes sign. The net
charge has its influence on the electric field, which in its turn determines the velocity of
the electrons: for low electric fields we have ohmic behaviour, for large electric fields the
velocity of the electrons will be saturated.
All these effects determine the effective resistance of the epilayer. As is well known,
the potential drop over the collector region can cause quasi-saturation. In that case the
external base-collector bias is in reverse, which is normal in forward operation, but the
internal junction is forward biased. Injection of holes into the epilayer then takes place.
The charge in the epilayer and in the base-collector region depends on the carrier concentrations in the epilayer, and will increase significantly in the case of quasi-saturation.
The electric field in the epilayer is directly related to the base-collector depletion capacitance. The avalanche current is determined by the same electric field, and in particular by
its maximum. Hence our description of the epilayer must also include a correct description of the electric field.
4.2 Some qualitative remarks on the description of the epilayer
Let us now concentrate on a one-dimensional model of the lightly doped epilayer. We
assume the epilayer to be along the x-axis from x = 0 to x = Wepi , as has been schematically shown in Fig. 7. The base is then located at x < 0, while the highly doped collector
region, the buried layer, is situated at x > Wepi . We assume a flat dope in the epilayer and
an abrupt epi-collector junction for the derivation of our equations. In the final description
of the model some factors have been generalised to account for non-ideal profiles. (Like,
©Koninklijke Philips Electronics N.V. 2002
27
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Usage of Mextram 504
Em. Base
epilayer
E1 B2 C2
Unclassified report
Collector
C1
0
Wepi
Figure 7: Schematic representation of the doping profile of a one-dimensional bipolar
transistor. One can observe the emitter, the base, the collector epilayer and the (buried)
collector. Constant doping profiles are assumed in many of the derivations. The collector
epilayer is in this chapter located between x = 0 and x = W epi and has a dope of N epi .
We have also shown where the various nodes E 1 , B2, C2 and C 1 of the intrinsic transistor
are located approximately.
for instance, depletion charges and capacitances that have a parameter for the grading
coefficient, instead of having the ideal grading coefficient of 1/2.) For the same reason
most of the parameters will have an effective value. This is even more so when current
spreading is taken into account.
We assume that the potential of the buried layer, at the interface with the epilayer, is given
by the node potential VC1 . The resistance in the buried layer and further away at the
collector contact are modelled by the resistance RCc and will not be discussed here.
We assume that the doping concentration in the base is much higher than that in the
epilayer. In that case the depletion region will be located almost only in the epilayer (i.e.
we have a one-sided pn-junction). The potential of the internal base (i.e. the base potential
while neglecting the base resistance) is given by VB2 .
Velocity saturation The drift velocity of carriers is given by the product of the mobility
and the electric field. The mobility of the electrons itself, however, also depends on the
electric field. It has a low field value µn0 , such that the low-field drift velocity equals
v = µn0 E. At high electric fields, however, this velocity saturates. the maximum value
given by the saturation velocity vsat . A simple equation that can be used to describe this
effect is
µn =
µn0
;
1 + µn0 E/vsat
v = µn E =
µn0 E
.
1 + µn0 E/vsat
(4.1)
As one can see there is a cross-over from v = µn0 E for small electric fields to v = vsat
for large electric fields. This cross-over happens at the critical electric field defined by
Ec =
vsat
.
µn0
(4.2)
Typical vales for Si are vsat = 1.07 · 107 cm/s, µn0 = 1.0 · 103 cm2 /Vs and E c =
7 · 103 V/cm.
28
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
4. The collector epilayer model
August 2002— NL-UR 2002/823
The current In normal forward mode electrons move from base to collector, i.e. in
positive x-direction. The current density Jepi is then negative, due to the negative charge
of electrons. The current Iepi itself, however, is generally defined as going from collector
to emitter, via the base, and is positive in forward mode. We therefore write
Iepi = − Aem Jepi .
(4.3)
The electric field As mentioned before, the electric field in the epilayer is important.
The basic description of the electric field is the same as that in a simple pn-junction. In the
epilayer (of an NPN) it is negative. According to general pn-junction theory, the integral
of the electric field from node B2 to node C1 equals the applied voltage VC1 B2 plus the
built-in voltage VdC :
Z Wepi
Z C1
E(x)dx = −
E(x)dx = VC1 B2 + VdC .
(4.4)
−
B2
0
Here we assumed that the electric field in the base and in the highly doped collector drops
very fast to zero, such that the contribution to the integral only comes from the region
0 < x < Wepi .
Equation (4.4) is an important limitation on the electric field. It is in itself not enough to
find the electric field. To this end we need Gauss’ law
ρ
dE
= .
(4.5)
dx
ε
Here ρ is the total charge density, given by
ρ = q(Nepi − n + p).
(4.6)
Consider now the electric field in an ohmic region. It is constant and has the value
E=
Iepi
Jepi
=−
.
σ
σ Aem
(4.7)
Here σ is the conductivity. The electric field is negative, as mentioned before. In ohmic
regions the electric field is low enough to prevent velocity saturation. The net charge is
zero and the number of electrons equals the dope Nepi . A negligible number of holes are
present. The ohmic resistance of the epilayer can then be calculated and is given by the
parameter RCv = Wepi /qµn Aem Nepi .
Next we consider the depletion regions. In these regions the electric field will be high.
Hence we can assume that the velocity of electrons is saturated. There will be no holes
in these regions either. The electron density however depends on the current density.
Since the electron velocity is constant we have n = |Jepi |/vsat . The total net charge
is then given by a sum of the dope and the charge density resulting from the current:
ρ = q Nepi − |Jepi |/vsat . For the charge density it does not matter whether the current
moves forth or back. This gives us
q Nepi
Iepi
dE
=
1−
,
(4.8)
dx
ε
Ihc
©Koninklijke Philips Electronics N.V. 2002
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E
Usage of Mextram 504
Wepi x
0
E
Unclassified report
Wepi x
0
I<Iqs
I<Iqs
I=Iqs
I~Ihc
I>Iqs
I=Iqs
I>Iqs
I=0
I=0
a)
b)
Figure 8: Figure describing the electric field in the epilayer as function of current. In a)
the width of the depletion layer decreases because ohmic voltage drop is the dominant
effect. In b) the width of the depletion layer increases because velocity saturation is
dominant (Kirk effect). At I = Iqs quasi-saturation starts (see text).
where we defined the hot-carrier current Ihc = q Nepi Aem vsat . When the epi-layer current
equals the hot-carrier current the total charge in that part of the epilayer will vanish. We
still call these regions depleted, since the electrons still move with vsat , in contrast to the
ohmic regions.
For currents larger than the hot-carrier current the derivative of the electric field will be
negative. There will still be a voltage drop over the epilayer. This voltage drop, however,
is no longer ohmic, but space-charge limited. The corresponding resistance of the epilayer
is now given by the Space-Charge Resistance SCRCv . We will discuss this in more detail
below.
Let us consider the current dependence of the electric field distribution in some more
detail, for both cases discussed above. At low current density (i.e. before quasi-saturation
defined below) the electric field in the epilayer is similar to that of a diode in reverse
bias. Next to the base we have a depletion region. This region is followed by an ohmic
region. When the current increases the width of the depletion layer changes. There are
two competing effects that make that this width either increases or decreases, which we
will discuss here quantitatively.
We know that the bias over the depletion region itself is given by the bias VC1 B2 minus the
ohmic potential drop. Hence when the ohmic region is large the intrinsic junction potential
will decrease with current, and so will the depletion region width. This is schematically
shown in Fig. 8a). At some point the depletion layer thickness vanishes, and the whole
electric field is used for the ohmic voltage drop. Since at higher currents we still need
to fulfil Eq. (4.4) the electric field becomes smaller close to the base. This is possible
because holes get injected into the epilayer, which reduces the resistance in the region
next to the base. This effect, quasi-saturation, will be discussed in more detail later.
The other effect that has an influence on the width of the electric field is velocity saturation. As can be seen from Eq. (4.8), the slope of the electric field decreases with increasing
current. This means that to keep the total integral over the electric field constant, as in
Eq. (4.4), the width of the depletion layer must increase. This is schematically shown
30
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
4. The collector epilayer model
August 2002— NL-UR 2002/823
in Fig. 8b). With increasing current the depletion width will continue to increase, until
it reaches the highly doped collector. For even higher currents the total epilayer will be
depleted. The slope of the electric field still decreases and can change sign. At some level
of current the value of the electric field at the base-epilayer junction drops beneath the
critical field E c for velocity saturation and holes get injected into the epilayer. As before,
at this point high injection effects in the epilayer start to play a role. This is again the
regime of quasi-saturation. When quasi-saturation is due to a voltage drop as a result of
the reversal of the slope of the electric field, the effect is better known as the Kirk effect.
Note that in both cases described above a situation occurs where the electric field is (approximately) flat over the whole epilayer, as shown in Fig. 8. In the ohmic case this will
happen at much smaller electric field (and therefore collector-base bias) than in the case
of space charge dominated resistance (Kirk effect).
Quasi-saturation Consider the normal forward operating regime. The (external) base
collector bias will be negative: VB2 C1 < 0. The epilayer, however, has some resistance,
which can either be ohmic, or space charge limited, as discussed above. As a result the
internal base-collector bias, in our model given by V B∗2C2 , is less negative than the external
bias. For large enough currents, it even becomes forward biased. This also means that the
carrier densities at the base-collector interface increase. At some point, to be more precise
when V B∗2C2 ' VdC , these carrier densities become comparable to the background doping.
From there on high-injection effects in the epilayer become important. This is the regime
of quasi-saturation. Note that we use the term quasi-saturation when the voltage drop is
due to an ohmic resistance, but also when it is due to a space-charge limited resistance, in
which case the effect is also known as Kirk effect.
For our description the current at which quasi-saturation starts, Iqs , is very important.
So let us consider it in more detail. As mentioned before, quasi-saturation starts when
V B∗2C2 = VdC . In that case we can express the integral over the electric field in terms of
Vqs , the potential drop over the epilayer, using Eq. (4.4):
Vqs = VdC − VB2 C1 = −
Z
Wepi
E(x)dx.
(4.9)
0
So, at the onset of quasi-saturation the integral over the electric field is fixed by the external base-collector bias, and does no longer depend on the current. We can then use
the relation between the electric field and the current to determine the current Iqs . In the
ohmic case the electric field is constant over the epilayer. The voltage drop is simply the
ohmic voltage drop and we can write
Iqs = Vqs /RCv .
(4.10)
For higher currents the electric field is no longer constant, due to the net charge present in
the epilayer. Its derivative is given by Eq. (4.8) and depends on the current. The current
at onset of quasi-saturation can still be given as Vqs over some effective resistance:
Iqs = Vqs /SCRCv .
©Koninklijke Philips Electronics N.V. 2002
(4.11)
31
NL-UR 2002/823— August 2002
Emitter
Usage of Mextram 504
Base
Unclassified report
Collector
Doping density
Electron density
Hole density
Collector
epilayer
−→ Nepi
0
xi
Wepi
Figure 9: Schematic view of the doping, electron and hole densities in the base-collector
region (on an arbitrary linear scale), in the case of base push-out/quasi-saturation. It also
shows the thickness of the epilayer Wepi and the injection layer x i . From Refs. [17, 18].
The effective resistance SCRCv is the space-charge resistance introduced above.
When the (internal) base-collector is forward biased, as in quasi-saturation, holes from
the base will be injected in the epilayer. Charge neutrality is maintained in this injection
layer, so also the electron density will increase. As we noted already in the description of
the main current, at high injection the hole and electron densities will have a linear profile.
This linear profile in the base is now continued into the epilayer. The width of the base
has effectively become wider, from the base-emitter junction to the end of the injection
region in the epilayer. This is known as base push-out and is shown in Fig. 9. It decreases
transistor performance considerably. As an example we show the output characteristics
in Fig. 10. Note that in the Spice-Gummel-Poon model quasi-saturation is not modelled.
The reduction of the current as modelled by Mextram shows the effect.
It is important to note that although the hole density profile and the electron density profile
are similar, only the electrons carry current. The electric field and the density gradient
work together to move the electrons. However for the holes they act opposite and create
an equilibrium. This equilibrium will be used to determine the electric field (which will
be considerably below the critical electric field E c ), as is being done in the Kull model.
Also the electron current in the injection region will in Mextram be described by the Kull
model.
32
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
4. The collector epilayer model
August 2002— NL-UR 2002/823
Spice-Gummel-Poon
IC (A)
0.02
Mextram
0.01
0.00
0
1
2
3
4
5
6
VCE (V)
Figure 10: The output characteristics for both the Spice-Gummel-Poon model and the
Mextram model. From Refs. [17, 18].
©Koninklijke Philips Electronics N.V. 2002
33
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Usage of Mextram 504
Unclassified report
5 The substrate network
Let us consider the substrate under and surrounding the transistor. Mextram models the
collector-substrate depletion capacitance. (We will disregard the substrate current here,
since it is negligible in normal forward operation.) This means that the Mextram external
substrate node is physically located just below the depletion layer in the substrate. This is
not the same location as where the substrate is connected to the ground at the level of the
interconnect. So there is a physical separation between the Mextram substrate node and
the substrate contact. For accurate modelling one needs at least a resistance between the
two locations or nodes, as in Fig. 11.
B
E
C
S
C cs
Mextram
S
R sub
Figure 11: The most simple substrate network consists of only a resistance between the
external Mextram node S and the substrate contact at the top of the wafer.
Mextram does not have this substrate resistance. Hence one should add it in a macro
model. The reason that Mextram does not have this substrate resistance inside the model
is related to the fact that this substrate resistance is not known up front. Its actual value
depends very much on the location of the substrate plug and substrate contact. Since this
location is determined by the designer, at the layout stage, this resistance can also only be
determined after the layout is known. Extracting the substrate resistance from on-wafer
measurements is possible, but it will not necessarily give the correct value in a real circuit.
By having it already as a parameter in the model, the designer will assume it is accurate,
which it is not.
There is another reason for not having the substrate resistance within the Mextram model.
The distance between the depletion layer of the substrate and the substrate contact is
quite large compared to the other transistor dimensions. This also means that at normal
operating frequencies non-quasi-static effects can already play a role. The most important
contribution to this is probably the charge storage in the AC substrate current path. For
this reason one often sees a substrate network as in Fig. 12.
In some cases it is not even accurate to assume a single non-distributed collector-substrate
depletion capacitance. In that case an even more elaborate substrate network is needed [19].
34
©Koninklijke Philips Electronics N.V. 2002
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5. The substrate network
B
E
August 2002— NL-UR 2002/823
C
Rplug
C cs
Mextram
S
S
R sub
C sub
Figure 12: The more complicated substrate network consists of a resistance R sub describing the resistance in the substrate itself, together with its semiconductor capacitance C sub
and an extra plug/contact resistance R plug .
So in practice the accuracy needed for the substrate network determines very much how
this substrate network should look like. For that reason all of the substrate network is kept
outside of Mextram.
©Koninklijke Philips Electronics N.V. 2002
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Usage of Mextram 504
Unclassified report
6 Usage of (self)-heating with Mextram
Transistors within a circuit will dissipate power. The generated power has an influence on
the temperature of the device and its surroundings. Hence, due to the power dissipation
devices in the circuit will get warmer. This is called (self)-heating. In Mextram there
are two ways to take this (self)-heating into account. The most direct and well-known
way is to include a self-heating network for each transistor and couple these networks to
each other via an external thermal network. In this way the heating will be taken into
account dynamically. The other way is to estimate the average static temperature increase
for each device and use the parameter DTA to increase the temperature of the device.
This method is much simpler both in implementation and in circuit simulation than using
heating networks. We will discuss both methods below. Note that both methods are
independent of each other: they do not interact.
6.1 Dynamic heating
Heating can be due to the transistor itself (self-heating) and due to other transistors (mutual heating). Here we discuss the heating if taken into account dynamically.
6.1.1 Self-heating
To describe self-heating we need to consider two things: what is the dissipated power and
what is the relation between the dissipated power and the increase in temperature.
Dissipated power The power that flows into a device can be calculated as a sum over
currents times voltage drops. For instance for a three-terminal bipolar transistor we can
write
P = IC VCE + I B VBE .
(6.1)
Since normally the collector current is larger than the base current and the collector voltage is larger than the base voltage, the first term is usually dominant.
Not all the power that flows into a transistor will be dissipated. Part of it will be stored as
the energy on a capacitor. This part can be released later on. So to calculate the dissipated
power, we need to add all the contributions of the dissipated elements, i.e., all the DC
currents times their voltage drops. We then get
X
Ibranch · Vbranch .
(6.2)
Pdiss =
all branches
The difference between the power flow into the transistor and the dissipated power as
calculated above, is stored in the capacitances.
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6. Usage of (self)-heating with Mextram
August 2002— NL-UR 2002/823
br
6
Æ
Æ
r
dT
Rth
Cth
Pdiss
Figure 13: The self-heating network. Note that for increased flexibility the node dT is
made available to the user (see also Fig. 14).
Relation between power and increase of temperature Next we need a relation between the dissipated power and the rise in temperature. In a DC case we can assume
a linear relation: 1T = Rth Pdiss , where the coefficient Rth is the thermal resistance
(in units K/W). We assume here that it does not matter where the power is actually dissipated. In reality, of course, a certain dissipation profile will also give a temperature profile
over the transistor: not every part will be equally hot. We will not take this into account.
In non-stationary situations we have to take the finite heat capacity of the device into
account [20]. So we must ask ourselves, what happens when a transistor is heated by a
small heat source. The dissipated power creates a flow of energy, driven by a temperature
gradient, from the transistor to some heat sink far away. The larger the gradient in the
temperature, the larger the flow. This means that locally the temperature in the transistor
will be larger than in the surrounding material. This increased temperature 1T is directly
related to the increase in the energy 1U :
1U = Cth 1T,
(6.3)
where Cth is the thermal capacitance (or effective heat capacitance) in units J/K. A part
of the dissipated power will now flow away, and a part will be used to increase the local
energy density if the situation is not yet stationary. Hence we can write
Pdiss =
1T
d1T
.
+ Cth
Rth
dt
(6.4)
Implementation For the implementation of self-heating an extra network is introduced,
see Fig. 13. It contains the thermal resistance Rth and capacitance Cth , both connected
between ground and the temperature node dT . The value of the voltage VdT at the temperature node gives the increase in local temperature. The power dissipation as given above
is implemented as a current source.
©Koninklijke Philips Electronics N.V. 2002
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b
r 6
Æ
Æ
r
transistor 1
Usage of Mextram 504
r
External network
Unclassified report
b
r 6
Æ
Æ
r
transistor 2
Figure 14: An example of mutual (and self-) heating of two transistors.
6.1.2 Mutual heating
Apart from self-heating it is also possible to model mutual heating of two or more transistors close together. To do this the terminals dT of the transistors have to be coupled
to each other with an external network. An example is given in Fig. 14. This external
network is not an electrical network, but a network of heat-flow and heat-storage (just
as the self-heating network within Mextram is not an electrical network). One has to be
careful, therefore, not to connect any ‘thermal’ nodes with ‘electrical’ nodes. The external
network can be made as complicated as one wishes, thermally connecting any number of
transistors. For more information we refer to literature, e.g. Refs. [21, 22, 23].
6.1.3 Advantages
The advantages of this method are that heating is taken into account dynamically. Hence
the temperature increase of a certain device depends on the actual power dissipation of the
device itself, as well as of its surroundings. Any time delays are also taken into account.
6.1.4 Disadvantages
There are a number of disadvantages. First of all, to be able to have an accurate description
of the temperature increase of a device one needs to have an accurate thermal network.
This means that one needs a lot of thermal resistances and thermal capacitances to give
an accurate result. In a sense, one must discretise the heat equations to get a good threedimensional heat flow. It is not easy to determine all these resistances and capacitances.
It is even not easy to determine the self-heating resistance of a single device in its circuit
surroundings, which differs from the surroundings in an on-wafer extraction module.
One of the other disadvantages has to do with the difference of time scale between heating effects (of the order of 1 µs or slower depending on the distance) and the time scale
of electronic signals (often of the order of 1 ns or less). This can cause difficulties in
simulations. It takes many signal cycles to generate a good number for the average power
dissipation.
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6. Usage of (self)-heating with Mextram
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Another important disadvantage is the extra increase in complexity and hence simulation
time. First of all, the extra thermal network has to be simulated. But this will, in practice
not be too bad. Then for all devices one must take into account all the derivatives w.r.t.
temperature. This takes extra time. And then the addition of self-heating makes the
convergence process of the simulation more difficult. It will therefore take more time.
And for difficult circuits (in the simulation sense) convergence might not even be reached.
6.2 Static heating
In practice it is often not necessary to take self-heating into account dynamically. Often
the circuit will have an average dissipation. This dissipation will generate a certain temperature profile over the circuit. For instance, a power intensive circuit will be hotter than
its surroundings. Changes in this temperature profile are much slower than the time-scales
on which the circuit operates. Only the time-averaged power dissipation is then relevant.
For these cases Mextram (and other Philips models) have an extra parameter DTA. This
parameter is used to increase the local ambient temperature w.r.t. the global ambient temperature. It is very useful for giving a part of the circuit an increased temperature. Since
many complex designs contain various circuit blocks, it is indeed quite handy to be able to
give the more power-intensive blocks a larger local ambient temperature. This feature of
using DTA to increase the local temperature can also be used for more complex situations,
where for instance the temperature has a gradient. This can occur, for instance, close to a
power-intensive circuit block.
6.2.1 Advantages
The method is very fast. Once the temperature profile is given and the parameters DTA
are set, the simulation does not take more time than a simulation which does not include
heating. Instances of Mextram transistors without self-heating (and hence without the
self-heating node) can be used.
6.2.2 Disadvantages
The method is less accurate. The increase in temperature depending on whether a transistor is on or not is not taken into account.
Of course one must have a method to estimate the temperature profiles. This is not always
easy. On the other hand, it is probably easier than making a complete thermal network.
6.3 Combining static and dynamic heating
As discussed above, there are two methods for increasing the temperature of the device.
The easiest method is using the parameter DTA to give a static heating. This increases the
©Koninklijke Philips Electronics N.V. 2002
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Usage of Mextram 504
Unclassified report
local ambient temperature:
Tlocal ambient = Tglobal ambient + DTA.
(6.5)
Apart from that, it is also possible to take dynamic heating into account, using the selfheating network, possibly connected to an external thermal network to give mutual heating. The device temperature is then given by
Tdevice = Tlocal ambient + (1T )dynamic heating .
(6.6)
It is important to realise that within Mextram these two ways of heating the transistor
are independent and that the temperature increases are additive. An increase using DTA
therefore does not generate a heat-flow through a thermal network.
It is perfectly possible to take self- and mutual heating of two critical transistors into
account, even if both have a, possibly different, local ambient temperature increase. Consider for instance the following situation. We have two transistors close together, like in
Fig. 14. One of them is also heated statically by a nearby part of the circuit and therefore
has some DTA > 0. We assume that the second transistor is further away from this heatsource. In that case the second transistor will be heated both by the static heat-source, as
well as by the first transistor. For the heating by the first transistor the thermal network
as in Fig. 14 takes care. In principle this same thermal network could be used to model
the heat flow due to the static heat-source from the first transistor to the second transistor.
In Mextram the parameter DTA does, however, not increase the value of 1T at the thermal node of the first transistor (because static and dynamic heating are independent). An
increased DTA at the first transistor therefore does not generate a heat-flow to the second
transistor. So the thermal network can not be used in conjunction with a static increase
via DTA. It is important to realise that this is not a mistake in the model. In some sense it
is even more physical, as we will show below. Interaction between both ways of heating
is not even needed. The independence between static and dynamic heating is therefore
rather a feature than a short-coming.
So let us consider the physical behaviour. The heat from the static heat-source is actually
not flowing to the first transistor, and from there on to the second transistor. It is flowing
from the source in all directions and generates a static temperature distribution. It can
flow around the first transistor. The value of the resulting temperature increase at the
location of the second transistor is the value that must be given to the DTA of this second
transistor. In this way not only the effects of heating by a static heat-source and dynamic
heating via a thermal network are independently added, also the thermal behaviour of both
heat-flows can be taken to be independent. For modelling two nearby transistors a very
simple thermal network, as in Fig. 14 is enough. This simple thermal network is, however,
insufficient to describe the heat-flow from the static source. For the static source, on the
other hand, one does not need to make such a thermal network, since only the temperature
profile is important.
Furthermore, it is not even necessary to take the static heating into account via a thermal
network. Once the thermal network is given, and once the temperature (or power dissipation) of the static source is know, the increase of temperature at each transistor in the
40
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
6. Usage of (self)-heating with Mextram
August 2002— NL-UR 2002/823
thermal network due solely to the static source can be calculated up front. The parameter
DTA can be used for this temperature increase. Any dynamic heating is then added by the
simulator. The only assumption in this is that the dynamic heating does not influence the
heat-flow due to the static source. The possible errors made by this assumption are far
less then the errors made by the lack of knowledge about actual temperature profiles or
heat flow.
6.4 The thermal capacitance
The reason for having a thermal capacitance was already explained above. In practice the
thermal capacitance is often not very important. For DC simulations is has no influence
at all. For RF simulations the delay time of the self-heating is so long that only the
average dissipated power is important, and not the instantaneous. The frequencies where
the thermal capacitance becomes important are around 1 MHz.
When you look at the small signal parameters of the transistor, for instance the output
conductance, then you see a change as function of frequency. For a SiGe transistor, for
instance, the collector current at fixed base current will decrease as function of collector
voltage due to self-heating. This means that the low-frequency output conductance is negative. At very high frequencies self-heating has no influence on the output conductance,
and it will be positive. As function of frequency one will therefore observe a change from
negative values to positive values. In the case of pure Si transistors the output conductance
will not be negative at low currents, but still one can see a transition.
Although the exact value of the thermal capacitance is often not important for simulations,
it is not good to give it a value of 0, say as a default in a parameter set. The reason for this
is that the RF simulations will give a wrong result (effectively, as if it were low-frequency
simulations).
©Koninklijke Philips Electronics N.V. 2002
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Usage of Mextram 504
Unclassified report
7 Operating point information
The operating point information is a list of quantities that describe the internal state of
the transistor. When a circuit simulator is able to provide these, it might help the designer
understand the behaviour of the transistor and the circuit.
The full list of operating point information consists of three parts. First all the branch
biases, the currents and the charges are present. Then there are the elements that can be
used if a full small-signal equivalent circuit is needed. These are all the derivatives of the
charges and currents. At last, and this is the most informative for a designer, the operating
point information gives usable approximations for use in a hybrid-π model. This hybridπ model is the basic model used by many designers for hand-calculations. It should give
similar results as Mextram, as long as neither the current, nor the frequency are too high.
In addition also the cut-off frequency is included in the operating point information.
7.1 Approximate small-signal circuit
In our presentation, we will start with the hybrid-π model. The approximate small-signal
model is shown Fig. 15. This model contains the following elements that can be found
from in operating point information (for the derivation of these various quantities, we refer to Ref. [11]):
gm
β
gout
gµ
RE
rB
RCc
CBE
C BC
CtS
Transconductance
Current amplification
Output conductance
Feedback transconductance
Emitter resistance
Base resistance
Constant collector resistance
Base-emitter capacitance
Base-collector capacitance
Collector-substrate capacitance
As mentioned before, one can also find the cut-off frequency of the transistor in the operating point information. The approximation for f T is much more accurate than can be
found from the equivalent circuit above.
fT
Good approximation for cut-off frequency
Related to self-heating, the dissipation and actual temperature are also available
Pdiss
TK
Dissipation
Actual temperature
Then there are a few extra quantities available for the experienced user:
Iqs
xi /Wepi
V B∗2C2
42
Current at onset of quasi-saturation
Thickness of injection layer
Physical value of internal base-collector bias
©Koninklijke Philips Electronics N.V. 2002
Unclassified report
B
r
r
7. Operating point information
rB
B2
CBE
β
gm
C BC
6
Æ
Æ
gµ v C 1 E 1
C1
?Æ
Æ
RCc
gm v B 2 E 1
E1
r
August 2002— NL-UR 2002/823
S Ct
S
r
C
1
gout
RE
E
Figure 15: Small-signal equivalent circuit describing the approximate behaviour of the
Mextram model. The actual forward Early voltage can be found as Veaf = IC /gout − VCE .
which can be different from the parameter value V ef , especially when dEg 6 = 0.
7.2 DC currents and charges
In this section the biases, DC currents and charges are listed.
Since we have 5 internal nodes we need 5 voltage differences to describe the bias at each
internal node, given the external biases. We take those that are the most informative for
the internal state of the transistor:
VB2 E1 Internal base-emitter bias
VB2 C2 Internal base-collector bias
VB2 C1 Internal base-collector bias including epilayer
VB1 C1 External base-collector bias without contact resistances
VE1 E Bias over emitter resistance
The actual currents (and charges, see next page) are:
IN
IC1 C2
I B1 B2
I B1
I BS1
I B2
I B3
Isub
Iavl
Iex
XIex
XIsub
ISf
I RE
I R Bc
I RCc
Main current
Epilayer current
Pinched-base current
Ideal forward base current
Ideal side-wall base current
Non-ideal forward base current
Non-ideal reverse base current
Substrate current
Avalanche current
Extrinsic reverse base current
Extrinsic reverse base current
Substrate current
Substrate failure current
Current through emitter resistance
Current through constant base resistance
Current through constant collector resistance
©Koninklijke Philips Electronics N.V. 2002
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NL-UR 2002/823— August 2002
QE
QtE
Q tSE
QBE
Q BC
Q tC
Q epi
Q B1 B2
Q tex
XQ tex
Q ex
XQ ex
Q tS
Usage of Mextram 504
Unclassified report
Emitter charge or emitter neutral charge
Base-emitter depletion charge
Sidewall base-emitter depletion charge
Base-emitter diffusion charge
Base-collector diffusion charge
Base-collector depletion charge
Epilayer diffusion charge
AC current crowding charge
Extrinsic base-collector depletion charge
Extrinsic base-collector depletion charge
Extrinsic base-collector diffusion charge
Extrinsic base-collector diffusion charge
Collector-substrate depletion charge
7.3 Elements of full small-signal circuit
The small-signal equivalent circuit contains the following conductances. In the terminology we use the notation A x , A y , and A z to denote derivatives of the quantity A to some
voltage difference. We use x for base-emitter biases, y is for derivatives w.r.t. VB2 C2 and
z is used for all other base-collector biases. The subindex π is used for base-emitter base
currents, µ is used for base-collector base currents, Rbv for derivatives of I B1 B2 and Rcv
for derivatives of IC1 C2 .
(See next page)
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©Koninklijke Philips Electronics N.V. 2002
Unclassified report
Quantity
gx
gy
gz
gπS
gπ,x
gπ,y
gπ,z
gµ,x
gµ,y
gµ,z
gµex
Xgµex
g Rcv,y
g Rcv,z
rbv
g Rbv,x
g Rbv,y
g Rbv,z
RE
R Bc
RCc
gS
Xg S
gSf
7. Operating point information
Equation
∂ I N /∂ VB2E1
∂ I N /∂ VB2C2
∂ I N /∂ VB2C1
∂ I BS1 /∂ VB1 E1
∂(I B1 + I B2 )/∂ VB2 E1
∂ I B1 /∂ VB2 C2
∂ I B1 /∂ VB2 C1
−∂ Iavl /∂ VB2E1
−∂ Iavl /∂ VB2C2
−∂ Iavl /∂ VB2C1
∂(Iex + I B3 /∂ VB1 C1
∂ XIex /∂ VBC1
∂ IC1 C2 /∂ VB2C2
∂ IC1 C2 /∂ VB2C1
1/(∂ I B1 B2 /∂ VB1 B2 )
∂ I B1 B2 /∂ VB2 E1
∂ I B1 B2 /∂ VB2 C2
∂ I B1 B2 /∂ VB2 C1
RET
RBcT
RCcT
∂ Isub /∂ VB1 C1
∂ XIsub /∂ VBC1
∂ ISf /∂ VSC1
August 2002— NL-UR 2002/823
Description
Forward transconductance
Reverse transconductance
Reverse transconductance
Conductance sidewall b-e junction
Conductance floor b-e junction
Early effect on recombination base current
Early effect on recombination base current
Early effect on avalanche current limiting
Conductance of avalanche current
Conductance of avalanche current
Conductance extrinsic b-c junction
Conductance extrinsic b-c junction
Conductance of epilayer current
Conductance of epilayer current
Base resistance
Early effect on base resistance
Early effect on base resistance
Early effect on base resistance
Emitter resistance (already given above)
Constant base resistance
Constant collector resistance (already given above)
Conductance parasitic PNP transistor
Conductance parasitic PNP transistor
Conductance substrate failure current
The small-signal equivalent circuit contains the following capacitances
Quantity
C BS E
C B E,x
C B E,y
C B E,z
C BC,x
C BC,y
C BC,z
C BCex
XC BCex
C B1 B2
C B1 B2,x
C B1 B2,y
C B1 B2,z
CtS
Equation
∂ Q tSE /∂ VB1 E1
∂(Q t E + Q B E + Q E )/∂ VB2E1
∂ Q B E /∂ VB2 C2
∂ Q B E /∂ VB2 C1
∂ Q BC /∂ VB2E1
∂(Q tC + Q BC + Q epi )/∂ VB2C2
∂(Q tC + Q BC + Q epi )/∂ VB2C1
∂(Q tex + Q ex )/∂ VB1 C1
∂(XQ tex + XQ ex )/∂ VBC1
∂ Q B1 B2 /∂ VB1B2
∂ Q B1 B2 /∂ VB2E1
∂ Q B1 B2 /∂ VB2C2
∂ Q B1 B2 /∂ VB2C1
∂ Q t S /∂ VSC1
©Koninklijke Philips Electronics N.V. 2002
Description
Capacitance sidewall b-e junction
Capacitance floor b-e junction
Early effect on b-e diffusion charge
Early effect on b-e diffusion charge
Early effect on b-c diffusion charge
Capacitance floor b-c junction
Capacitance floor b-c junction
Capacitance extrinsic b-c junction
Capacitance extrinsic b-c junction
Capacitance AC current crowding
Cross-capacitance AC current crowding
Cross-capacitance AC current crowding
Cross-capacitance AC current crowding
Capacitance s-c junction (already given above)
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©Koninklijke Philips Electronics N.V. 2002
Unclassified report
References
August 2002— NL-UR 2002/823
References
[1] For the most recent model descriptions, source code, and documentation, see the
web-site http://www.semiconductors.philips.com/Philips Models.
[2] J. C. J. Paasschens and W. J. Kloosterman, “The Mextram bipolar transistor model,
level 504,” Unclassified Report NL-UR 2000/811, Philips Nat.Lab., 2000. See
Ref. [1].
[3] J. J. Ebers and J. L. Moll, “Large signal behaviour of junction transistors,” Proc.
IRE, vol. 42, p. 1761, 1954.
[4] H. K. Gummel and H. C. Poon, “An integral charge control model of bipolar transistors,” Bell Sys. Techn. J., vol. May-June, pp. 827–852, 1970.
[5] R. S. Muller and T. I. Kamins, Device electronics for integrated circuits. Wiley, New
York, 2nd ed., 1986.
[6] S. M. Sze, Physics of Semiconductor Devices. Wiley, New York, 2nd ed., 1981.
[7] I. E. Getreu, Modeling the bipolar transistor. Elsevier Sc. Publ. Comp., Amsterdam,
1978.
[8] H. C. de Graaff and F. M. Klaassen, Compact transistor modelling for circuit design.
Springer-Verlag, Wien, 1990.
[9] J. Berkner, Kompaktmodelle für Bipolartransistoren. Praxis der Modellierung, Messung und Parameterbestimmung — SGP, VBIC, HICUM und MEXTRAM (Compact
models for bipolar transistors. Practice of modelling, measurement and parameter
extraction — SGP, VBIC, HICUM und MEXTRAM). Expert Verlag, Renningen,
2002. (In German).
[10] P. A. H. Hart, Bipolar and bipolar-mos integration. Elsevier, Amsterdam, 1994.
[11] J. C. J. Paasschens, W. J. Kloosterman, and R. van der Toorn, “Model derivation of Mextram 504. The physics behind the model,” Unclassified Report NL-UR
2002/806, Philips Nat.Lab., 2002. See Ref. [1].
[12] J. C. J. Paasschens, W. J. Kloosterman, and R. J. Havens, “Parameter extraction
for the bipolar transistor model Mextram, level 504,” Unclassified Report NL-UR
2001/801, Philips Nat.Lab., 2001. See Ref. [1].
[13] H. K. Gummel, “A charge control relation for bipolar transistors,” Bell Sys. Techn.
J., vol. January, pp. 115–120, 1970.
[14] J. C. J. Paasschens, W. J. Kloosterman, and R. J. Havens, “Modelling two SiGe
HBT specific features for circuit simulation,” in Proc. of the Bipolar Circuits and
Technology Meeting, pp. 38–41, 2001. Paper 2.2.
©Koninklijke Philips Electronics N.V. 2002
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[15] J. R. Hauser, “The effects of distributed base potential on emitter-current injection
density and effective base resistance for stripe transistor geometries,” IEEE Trans.
Elec. Dev., vol. May, pp. 238–242, 1964.
[16] H. Groendijk, “Modeling base crowding in a bipolar transistor,” IEEE Trans. Elec.
Dev., vol. ED-20, pp. 329–330, 1973.
[17] J. C. J. Paasschens, W. J. Kloosterman, R. J. Havens, and H. C. de Graaff, “Improved
compact modeling of ouput conductance and cutoff frequency of bipolar transistors,”
IEEE J. of Solid-State Circuits, vol. 36, pp. 1390–1398, 2001.
[18] J. C. J. Paasschens, W. J. Kloosterman, R. J. Havens, and H. C. de Graaff, “Improved modeling of ouput conductance and cut-off frequency of bipolar transistors,”
in Proc. of the Bipolar Circuits and Technology Meeting, pp. 62–65, 2000. Paper
3.3.
[19] S. D. Harker, R. J. Havens, J. C. J. Paasschens, D. Szmyd, L. F. Tiemeijer, and
E. F. Weagel, “An S-parameter technique for substrate resistance characterization of
RF bipolar transistors,” in Proc. of the Bipolar Circuits and Technology Meeting,
pp. 176–179, 2000. Paper 10.2.
[20] C. Kittel and H. Kroemer, Thermal Physics. Freeman & Co., second ed., 1980.
[21] J. S. Brodsky, R. M. Fox, and D. T. Zweidinger, “A physics-based dynamic thermal impedance model for vertical bipolar transistors on soi substrates,” IEEE Trans.
Elec. Dev., vol. 46, pp. 2333–2339, 1999.
[22] P. Palestri, A. Pacelli, and M. Mastrapasqua, “Thermal resistance in Si1−x Gex HBTs
on bulk-Si and SOI substrates,” in Proc. of the Bipolar Circuits and Technology
Meeting, pp. 98–101, 2001. Paper 6.1.
[23] D. J. Walkey, T. J. Smy, R. G. Dickson, J. S. Brodsky, D. T. Zweidinger, and R. M.
Fox, “A VCVS-based equivalent circuit model for static substrate thermal coupling,”
in Proc. of the Bipolar Circuits and Technology Meeting, pp. 102–105, 2001. Paper 6.2.
48
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